Introduction:
In this problem, we are given that a triangle has integral sides and the length of two sides are 2014 and 2015. We have to find the number of such triangles.

Approach:
To solve this problem, we can use the triangle inequality which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Using this inequality, we can say that the third side must satisfy the following condition:

2015 - 2014 < third="" side="" />< 2015="" +="" />

Simplifying this, we get:

1 < third="" side="" />< />

We know that the sides of the triangle are integers, so we need to count the number of integers between 1 and 4029 (inclusive) that can be the length of the third side of the triangle.

Solution:
To count the number of such integers, we can subtract the number of integers that are not valid from the total number of integers between 1 and 4029.

The total number of integers between 1 and 4029 (inclusive) is 4029.

The number of integers that are not valid can be calculated as follows:

- There are no integers less than or equal to 1 that can be the length of the third side.
- There are no integers greater than or equal to 4029 that can be the length of the third side.
- For any integer n between 2016 and 4028 (inclusive), the sum of 2014 and 2015 is greater than n, so n cannot be the length of the third side.

Therefore, the number of integers that are not valid is:

0 + 0 + (4028 - 2015 + 1) = 1014

So, the number of valid integers is:

4029 - 1014 = 3015

Therefore, the number of such triangles is 3015.

Conclusion:
In conclusion, we have used the triangle inequality to find that the third side of the triangle must be an integer between 1 and 4029 (inclusive). We have then counted the number of integers that are not valid and subtracted it from the total number of integers between 1 and 4029 to get the number of valid integers. Finally, we have concluded that the number of such triangles is 3015.